The British Met. Office Archives in Exeter (England) have both the British and the French meteorological records for the Great War. The published RFC / RAF Communiques also contain a small daily weather digest from July 1916 onwards (the OFF weather database is based largely on the latter, and is therefore somewhat 'generic', not differentiating between the different sectors of the Front).
Met Office: Library information
At the outbreak of the war there was no military meteorological service, and weather forecasting as a science was still in its infancy. But as the military situation in France and Flanders settled into its pattern of opposing trench systems, there was a growing need on both sides to predict short-term weather patterns There is little information available in English on either French or German weather forcasting during the war, or on their meteorological service's contribution to the war effort on the Western Front. It is known that the French Meteorological Service already had a well-established network of weather observation stations, and prepared simple daily plotted charts and weather summaries based on daily observations from these stations and on 3-hourly observations at Paris. These French weather charts and summaries are still available today, for the whole of the war period, from the Britsh Met. Office archives in Exeter (England), and presumbly from the French Met. Service archives also. It is very likely that the German Army and Air Service also had access to its own meteriological information, although if there are any surviving records relating to this I am not aware of them.
The British Met. Office had access to the French meteorological data, to the data from British weather stations and ships, and from its links to weather organisations in Spain and Scandinavia. This was enough to prepare forcasts based on larger scale weather patterns, such as frontal systems (the term 'weather front' was itself coined at around this time), but was not fine enough to predict local changes in weather along the Western Front. It also provided little information on changes to wind strength and direction above ground level (needed, increasingly, not only by the artillery but also by the Flying Corps and the anti-aircraft units along the Front). Initially, however, the requirement was for accurate and timely predictions of wind strength and direction at ground level, both along the Front as a whole and in specific sectors, to help the Army prepare against the new threat of gas attack from the German lines, and also to support the Engineers by predicting favourable local conditions for British gas attacks against the Germans. In June 1915, therefore, after the first German gas attack at Ypres, two Met. officers went to France to form the Meteorological Field Service, GHQ - 'Meteor' as it was soon to become known. Observers were recruited, mostly from the Artists Rifles. The new service was successful in accurately predicting local wind conditions for Allied gas attacks during the Battle of Loos, and was thereafter established as a section of the Royal Engineers to support RE gas companies. Although observations of surface winds formed the bulk of this early work, both the artillery and the RFC also needed infomation that the Met. observers could provide. The artillery needed information on temperature, wind strength and direction up to 2,000 ft (later to 6,000 ft), to be able to predict the 'drift' caused by wind on artillery shells. The RFC needed information on both wind and cloud formations at altitudes above this - as, at an early meeting with Met. officers at RFC HQ in France, General Henderson remarked that air reconnaisance had to be conducted above 3000 ft because "if they fly lower they are shot down like rabbits".
Met. officers were recruited and asigned to Army HQs, establishing a network of weather observation stations along the Front. As the war progressed, the emphasis changed away from gas attacks and towards closer support and integration - first with the artillery, and then increasingly with the RFC and the Independent Air Foce operating from Nancy. Small Met. sections were also sent with RFC detachments to the Dardenelles, Salonika and Italy. The night bombers also required up to date information on the 'sleep winds', the prevailing upper winds that were formed during the hours of darkness. Small weather pilot balloons were used at first, along with sound-ranging and observations of anti-aircraft bursts at different altitudes, and a kite balloon was also obtained for collecting weather observations. But from early 1918 a dedicated meteorological aircraft section was formed to take regular observations of the wind, humidity and temperature in the upper air up to 14,000 ft. These observations are said to have been especially useful before the opening of the final offensive on 8th August. From May 1917 weather observations were also being taken at Calais, initially four times a day, but rising to every two hours by the end of the war. By the end of the war the new Met. Service was producing daily weather reports (these, along with the reports from Calais, are also retained at the Met. Office archives in Exeter) and had expanded from the initial two officers in June 1915 to a final establishment of 28 officers and 187 other ranks. It had become an integral part of the organization of the RAF - subsequently rising to 750 officers and men by 1939, and nearly 6,800 by 1945.
Sources
Gold, E. Meteorology in the First World War. Weather, vol.5 no.9, 1950.
Gold, E. The Meteorological Office and the First World War. Meteorological Magazine, vol.84 pt.996, 1955.
Stagg, J.M. The Meteorologic Office and the Second World War. Meteorological Magazine, vol.84 pt.996, 1955.
Personal correspondence with Ian MacGregor, Archive Information Manager, Met. Office Archive
Bletchley
Edit: Unit diaries are also a very good source for very general weather observations for a particular location (but not often detailed, just "very wet" or "foggy in the morning" type entries). These are a very common feature of the British unit diaries I have seen, that suggests that there was a section on the form that invited weather related comments - so it may also be a feature of AEF diary entries.